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9

Put simply, and without any mathematical symbols, prior means initial beliefs about an event in terms of probability distribution. You then set up an experiment and get some data, and then "update" your belief (and hence the probability distribution) according to the outcome of the experiment, (the posteriori probability distribution). Example: Assume we ...


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A DBN means that your model is replicated over each time discrete time step t, called slice. Variables of each slice can be connected together, as well a from previous slice to a latter slice (in this precise direction only). The probability tables and links remain the same in each slice if the model is stationary, which is the default case (otherwise they ...


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Yes, ABC is a specific application of Monte Carlo method. That application is approximating likelihood functions. Anything that happens in a computer is actually deterministic. However, ABC, like any other MCM, must have a good pseudo-random generator. There is no difference in the amount of randomness required by the methods. Examples of Monte Carlo ...


2

This is a non-mathematical riddle, so part of the solution is determining the rules of the game. The computer science way of solving this, however, would be to use a Gray code and a synchronized system. The idea is as follows. Suppose that there were only 3 switches. You will go over their states in the following order: $$ 000\\001\\011\\010\\110\\111\\101\\...


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The correct formula is $$P(w|\text{ending}) = \frac{\text{count}(w,\text{ending}) + 1}{\text{count}(w) + N},$$ where $N$ is the number of possible values of $w$. Here $w$ ranges over the set of all words that you'll ever want to estimate $P(w|\text{ending})$ for: this includes all the words in the training text, as well as any other words you might want ...


2

An active learning approach using which combines an incrementally-learned Regression Tree with bandit-style sampling from leaf nodes to determine which instance to request a label for next is described in the “Adapting to Concept Drift in Credit Card Transaction Data Streams Using Contextual Bandits and Decision Trees” paper (disclaimer: I'm an author on the ...


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Mathematics and computer science doesn't have anything to say about whether simpler hypotheses are more likely. That's a question about reality, not about math / computer science. What computer science can provide is the notion of Kolmogorov complexity. Kolmogorov complexity is one reasonable notion of simplicity: bit-strings with lower Kolmogorov ...


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Create an extra class whose membership means "belongs to none of the predefined classes".


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This isn't a hidden Markov model; this is an ordinary Markov model. Take a look at Wikipedia's article on Markov chains and specifically the notion of a steady-state distribution (or stationary distribution), or read about the subject in your favorite textbook -- there are many that cover Markov chains.


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I don't think they map onto each other, aside from RHN's brief discussion of thalamo-cortical loops, the inspiration for SLT's so-called "Creativity Machines." In SLT's architectures, compelling/cogent solution patterns are sought within a stream of novel activation patterns (i.e., confabulations) driven by various forms of internal disturbances to ...


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The basic approach is to model the noise as an appropriate stochastic process (aka random process). There's an entire subfield of statistics that has studied different kinds of stochastic processes, and they each have different properties. It would be too much to try to summarize that entire subfield in a single answer here.


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Bayesian networks encode a factorization of the joint probability distribution of a set of variables. Specifically, each variable is conditionally independent of all its non-descendants given its parents. The joint probability distribution can be written as: $P(X) = \prod_i P(X_i|Pa(X_i))$ where Pa(X) are the parents of X in the network. You should be ...


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In a Bayesian network, a variable is independent from all the variables given its Markov blanket (except of course the variables in the Markov blanket). However, the Markov blanket is not the minimal set that renders two variables independent. Also note that a variable may be independent of some variables in the Markov blanket, given another set of ...


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